First of all, do NOT underestimate the power of the technically unskilled call center.
They might not be able to solve anything, but user satisfaction (which is a key success factor for the Service Desk) does not solely reply on the 1st line resolution rate. Users can accept (to a certain degree) that the operator is not able to solve their problem, but they can't accept patronizing operators, or operators using incomprehensible tech lingo.
Operators with both technical, empathical, and communicative skills are very rare. In my company we have a Service Desk staff with very strong technical skills, and most of them are absolutely useless when it comes to communicating with end users.
Our solution is a two-layer Service Desk. Layer 1 takes calls from end users and dispatches to layer 2 what they are not able to fix themselves. Layer 2 catches the balls forwarded from layer 1 AND takes calls from "super users".
This started working quite well when we forced the two layers to cooperate instead of fighting each other. Means to this end: Weekly manager meetings, exchange of staff.
I suggest you create a Layer 2 Service Desk within your own organization and give this function the full responsibility for end user support. They should be the owners of the Call Center outsourcing contract.
By having a Layer 2 Service Desk, the Call Center will have one and only one address for routing tickets. You will have a function with 1st hand knowledge of the efficiency of the Call Center. And you will have a function with immediate interest in handing over "FAQ-solutions" to the Call Center.

But can problem and change management survive without proper Incident management ?

your "incident management" hat does not need to be worn by your ticket takers. You can provide requirements for reporting that allow an analyst to perform the incident management function outside of the help desk and grow problem management over time.

Change Management can also live outside of the "help desk" too. It is not the best method but ticket takers can note examples against a change without being too involved in the CM process.

As long as your process is strong and sustainment teams subscribe/understand it, you can practice the model without tools while maintaining a low skilled desk.

Remember that best practice is only "best" until a model is implemented better suited to your business's need._________________"When music and courtesy are better understood and appreciated, there will be no war" - Confucius